Mumbai

Since you are doing B.Sc through lateral entry system, surely you can take admission in M.Sc in IT by lateral entry system. That would not need the problem. I advise you to just check the colleges like University of Mumbai for this admission. It's a very good college for computer science.

There are around 88 BBA colleges in Mumbai. Check the complete list of BBA colleges in Mumbai in given link:
https://www.shiksha.com/bba/colleges/bba-colleges-Mumbai-all

After 12th you can do BBA, B.Com or BBM. These all courses are available for you after 12th. You can also take admission for law.

There are over 13 Distance/Correspondence M.Sc. In Science colleges in Mumbai. Please refer to the link provided below for a detailed list of colleges which offers M.Sc. Courses via distance/correspondence. You should be able to navigate to a college of your choice using the filters on the web page. https://www.shiksha.com/science/colleges/distance-correspondence-m-sc-colleges-mumbai-all.

Ye, the choice of specialization is made in the 2nd year in the last semester.